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Properties

Label 4-155952-1.1-c1e2-0-8
Degree $4$
Conductor $155952$
Sign $-1$
Analytic cond. $9.94363$
Root an. cond. $1.77576$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 6-s − 8·7-s + 3·8-s + 9-s + 12-s + 8·14-s − 16-s − 18-s + 8·21-s − 3·24-s + 2·25-s − 27-s + 8·28-s − 5·32-s − 36-s − 8·41-s − 8·42-s + 16·43-s + 48-s + 34·49-s − 2·50-s + 8·53-s + 54-s − 24·56-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 3.02·7-s + 1.06·8-s + 1/3·9-s + 0.288·12-s + 2.13·14-s − 1/4·16-s − 0.235·18-s + 1.74·21-s − 0.612·24-s + 2/5·25-s − 0.192·27-s + 1.51·28-s − 0.883·32-s − 1/6·36-s − 1.24·41-s − 1.23·42-s + 2.43·43-s + 0.144·48-s + 34/7·49-s − 0.282·50-s + 1.09·53-s + 0.136·54-s − 3.20·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(155952\)    =    \(2^{4} \cdot 3^{3} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(9.94363\)
Root analytic conductor: \(1.77576\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 155952,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.i_dq
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.aq_fe
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.47.a_as
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ai_w
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ae_eg
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.ai_o
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ae_di
79$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.79.a_aco
83$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.83.a_ew
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.97.a_as
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.293302975859092155675164639167, −8.711389231997330910658962751364, −8.191461105185642051785958502675, −7.34795374339299502296346769124, −7.13957592611424172860801736774, −6.50153594331278040059111005849, −6.24836935840109374091271684275, −5.57694678007027267506208037636, −5.10299402705696517423407223963, −4.12642868947818347648540503072, −3.80884471379299016715244795860, −3.12632364494016894108562620356, −2.40402593173024140840610616688, −0.912048492770576860857164841091, 0, 0.912048492770576860857164841091, 2.40402593173024140840610616688, 3.12632364494016894108562620356, 3.80884471379299016715244795860, 4.12642868947818347648540503072, 5.10299402705696517423407223963, 5.57694678007027267506208037636, 6.24836935840109374091271684275, 6.50153594331278040059111005849, 7.13957592611424172860801736774, 7.34795374339299502296346769124, 8.191461105185642051785958502675, 8.711389231997330910658962751364, 9.293302975859092155675164639167

Graph of the $Z$-function along the critical line